Inmathematics, ameasurable space orBorel space^[1] is a basic object inmeasure theory. It consists of aset and aσ-algebra, which defines thesubsets that will be measured.

It captures and generalises intuitive notions such as length, area, and volume with a set${\displaystyle X}$ of 'points' in the space, butregions of the space are the elements of theσ-algebra, since the intuitive measures are not usually defined for points. The algebra also captures the relationships that might be expected of regions: that a region can be defined as an intersection of other regions, a union of other regions, or the space with the exception of another region.

Consider a set${\displaystyle X}$ and aσ-algebra${\displaystyle \mathcal{F}}$ on${\displaystyle X.}$ Then the tuple${\displaystyle (X,\mathcal{F})}$ is called a measurable space.^[2] The elements of${\displaystyle \mathcal{F}}$ are calledmeasurable sets within the measurable space.

Note that in contrast to ameasure space, nomeasure is needed for a measurable space.

Look at the set:$${\displaystyle X=\{1,2,3\}.}$$One possible${\displaystyle \sigma }$-algebra would be:$${\displaystyle {\mathcal{F}}_1=\{X,\varnothing \}.}$$Then${\displaystyle \left(X,{\mathcal{F}}_1\right)}$ is a measurable space. Another possible${\displaystyle \sigma }$-algebra would be thepower set on${\displaystyle X}$:$${\displaystyle {\mathcal{F}}_2=\mathcal{P}(X).}$$With this, a second measurable space on the set${\displaystyle X}$ is given by${\displaystyle \left(X,{\mathcal{F}}_2\right).}$

If${\displaystyle X}$ is finite or countably infinite, the${\displaystyle \sigma }$-algebra is most often thepower set on${\displaystyle X,}$ so${\displaystyle \mathcal{F}=\mathcal{P}(X).}$ This leads to the measurable space${\displaystyle (X,\mathcal{P}(X)).}$

If${\displaystyle X}$ is atopological space, the${\displaystyle \sigma }$-algebra is most commonly theBorel${\displaystyle \sigma }$-algebra${\displaystyle \mathcal{B},}$ so${\displaystyle \mathcal{F}=\mathcal{B}(X).}$ This leads to the measurable space${\displaystyle (X,\mathcal{B}(X))}$ that is common for all topological spaces such as the real numbers${\displaystyle \mathbb{R}.}$

The term Borel space is used for different types of measurable spaces. It can refer to

• any measurable space, so it is a synonym for a measurable space as defined above^[1]
• a measurable space that isBorel isomorphic to a measurable subset of the real numbers (again with the Borel${\displaystyle \sigma }$-algebra)^[3]

Families${\displaystyle \mathcal{F}}$ of sets over${\displaystyle \mathrm{\Omega }}$ v t e
Is necessarily true of${\displaystyle \mathcal{F}:}$ or, is${\displaystyle \mathcal{F}}$closed under:	Directed by${\displaystyle \supseteq }$	${\displaystyle A\cap B}$	${\displaystyle A\cup B}$	${\displaystyle B\setminus A}$	${\displaystyle \mathrm{\Omega }\setminus A}$	${\displaystyle A_1\cap A_2\cap \cdots }$	${\displaystyle A_1\cup A_2\cup \cdots }$	${\displaystyle \mathrm{\Omega }\in \mathcal{F}}$	${\displaystyle \varnothing \in \mathcal{F}}$	F.I.P.
π-system
Semiring										Never
Semialgebra(Semifield)										Never
Monotone class						only if${\displaystyle A_i\searrow }$	only if${\displaystyle A_i\nearrow }$
𝜆-system(Dynkin System)				only if ${\displaystyle A\subseteq B}$			only if${\displaystyle A_i\nearrow }$ or they aredisjoint			Never
Ring(Order theory)
Ring(Measure theory)										Never
δ-Ring										Never
𝜎-Ring										Never
Algebra(Field)										Never
𝜎-Algebra(𝜎-Field)										Never
Dual ideal
Filter				Never	Never				${\displaystyle \varnothing \notin \mathcal{F}}$
Prefilter(Filter base)				Never	Never				${\displaystyle \varnothing \notin \mathcal{F}}$
Filter subbase				Never	Never				${\displaystyle \varnothing \notin \mathcal{F}}$
Open Topology							(even arbitrary${\displaystyle \cup }$)			Never
Closed Topology						(even arbitrary${\displaystyle \cap }$)				Never
Is necessarily true of${\displaystyle \mathcal{F}:}$ or, is${\displaystyle \mathcal{F}}$closed under:	directed downward	finite intersections	finite unions	relative complements	complements in${\displaystyle \mathrm{\Omega }}$	countable intersections	countable unions	contains${\displaystyle \mathrm{\Omega }}$	contains${\displaystyle \varnothing }$	Finite Intersection Property
Additionally, asemiring is aπ-system where every complement${\displaystyle B\setminus A}$ is equal to a finitedisjoint union of sets in${\displaystyle \mathcal{F}.}$ Asemialgebra is a semiring where every complement${\displaystyle \mathrm{\Omega }\setminus A}$ is equal to a finitedisjoint union of sets in${\displaystyle \mathcal{F}.}$ ${\displaystyle A,B,A_1,A_2,\dots }$ are arbitrary elements of${\displaystyle \mathcal{F}}$ and it is assumed that${\displaystyle \mathcal{F}\ne \varnothing .}$

• Borel set – Class of mathematical sets
• Measurable function – Kind of mathematical function
• Measure – Generalization of mass, length, area and volume
• Standard Borel space – Mathematical construction in topology
• Category of measurable spaces

• Measure theory
• Space (mathematics)

• Articles with short description
• Short description is different from Wikidata